The Basic Equations of Astrophysics

    In order to adequately express and treat the physical conditions at a point in space, whether it be in a stellar core, atmosphere, or in the interstellar medium, we need a number of basic relationships (equations) that enable us to relate the basic variables.  Six relationships or types of relationships that are commonly used are the following:

(1) Equation of State - This is generally a function relating pressure, density and temperature,  f(P, ρ, T).  Often the perfect gas law, usually expressed P = (k/μH)ρT in astrophysics, is adequate.  However in the cores of many stars or throughout the entire volumes of white dwarfs or neutron stars, where degeneracy can be substantial, or in the interstellar medium where T is poorly defined, the perfect gas law is inadequate.  Also in very hot stars where radiation pressure is substantial, the perfect gas law is a poor approximation.

(2) The specific intensity of radiation - In its most general form I_ν(r, θ, φ) ≡ energy per unit time per unit area per unit frequency interval per unit solid angle in direction (θ, φ) at position r.  In cgs units [I_ν(r, θ, φ)] = ergs s^-1 cm^-2 Hz^-1 sterad^-1.  Alternatively the intensity may be expressed per unit wavelength, I_λ, rather than frequency interval, I_ν, but note that, since I_λdλ = I_νdν, I_λ = (c/λ²) I_ν = (ν²/c) I_ν.
        In thermodynamic equilibrium the radiation field is expressed by the Planck function, I_ν (θ, φ) = B_ν (T) = (2hν³/c²) / (e^hν/kT - 1).  Note that
B_ν(T) is isotropic.  In most situations I_ν (θ, φ) must be evaluated numerically and cannot be expressed analytically (although in some cases analytic approximations represent it fairly well).  It is evaluated by integration of a differential equation known as the equation of transfer.

(3) The velocity distribution of each kind of particle present with respect to some reference frame - In thermodynamic equilibrium this is expressed by the so-called Maxwellian distribution.

(4) The relative population of each ionization state for all species of atoms and molecules present.  This is expressed by the ionization (Saha) equation.

(5) The relative population of the ground and each excited state for every species of neutral atom, ion, neutral molecule and ionized molecule present.  This is expressed by the excitation (Boltzmann) equation.

(6) The relative numbers of atoms and molecules in every reacting assembly present, e.g.,  N(O)/N(OH) and N(H)/N(OH).  These ratios are expressed by the so-called dissociation equation.